Question: Simplify and expand the following expression: $ \dfrac{x - 8}{5x + 7}-\dfrac{x}{5x - 7} $
In order to subtract expressions, they must have a common denominator. Get both fractions over a common denominator of $(5x + 7)(5x - 7)$ Multiply the first term by $\dfrac{5x - 7}{5x - 7}$ $ \begin{align*} \dfrac{x - 8}{5x + 7} \times \dfrac{5x - 7}{5x - 7} & = \dfrac{(x - 8)(5x - 7)}{(5x + 7)(5x - 7)} \\ & = \dfrac{5x^2 - 47x + 56}{(5x + 7)(5x - 7)}\end{align*} $ Multiply the second term by $\dfrac{5x + 7}{5x + 7}$ $ \begin{align*} \dfrac{x}{5x - 7} \times \dfrac{5x + 7}{5x + 7} & = \dfrac{(x)(5x + 7)}{(5x - 7)(5x + 7)} \\ & = \dfrac{5x^2 + 7x}{(5x - 7)(5x + 7)}\end{align*} $ Now we have: $ = \dfrac{5x^2 - 47x + 56}{(5x + 7)(5x - 7)} - \dfrac{5x^2 + 7x}{(5x - 7)(5x + 7)} $ Now both terms have a common denominator we can subtract the numerators: $ = \dfrac{5x^2 - 47x + 56 - (5x^2 + 7x)}{(5x + 7)(5x - 7)} $ $ = \dfrac{5x^2 - 47x + 56 - 5x^2 - 7x}{(5x + 7)(5x - 7)} $ $ = \dfrac{-54x + 56}{(5x + 7)(5x - 7)}$ Expand the denominator: $ = \dfrac{-54x + 56}{25x^2 - 49}$